Mass Generation from Higgs-like Ghosts Scott Chapman Chapman University, One University Drive, Orange, CA 92866 (Dated: November 1, 2008) Abstract Covariant quantization of gauge theories generally requires the inclusion of Fadeev Popov ghosts in the gauge-fixed Lagrangian. Normally these ghosts have fermionic statistics, but in supersymmetric theories that include fermionic gauge fields, there can also be ghosts with bosonic statistics. Since these ghosts are scalar bosons, they can have vacuum expectation values (VEVs) without violating Lorentz invariance. In particular, for the supersymmetric group SU(2/3), one can choose a gauge with a Higgs-like bosonic ghost potential that is minimized when one of the ghosts develops a VEV. This VEV breaks the SU(2) x U(1) subgroup of SU(2/3) down to U(1) and spontaneously generates vector boson masses, but without the “hierarchy” problems that accompany the Higgs field. SU(2/3) also has an SU(3) subgroup, and unitarity requires that the SU(3) gauge bosons must be confined. Since bosonic ghosts do not exist as particles asymptotically, this kind of theory could be a possible explanation if no Higgs Boson is detected at the LHC. PACS numbers: 12.10.-g, 12.60.Jv Introduction Over the last thirty years, the Standard Model has been phenomenally successful in reproducing results from particle physics experiments and in correctly predicting the existence and properties of new particles. In fact, the only particle predicted by the Standard Model that has yet to be detected is the Higgs Boson, the scalar field needed to spontaneously generate masses for all other particles. However, in addition to a lack of direct experimental evidence, there are theoretical “hierarchy” problems associated with the “unnatural” [1] fine tuning required to renormalize the quadratic divergences of the Higgs field [2]. This unnatural fine tuning must be more precise than 1% if the Standard Model is used to describe physics at energy levels of 20 TeV or higher [3]. But even for energies as low as 1 TeV, it has been argued that the Higgs field makes the Standard Model “unnatural” [4]. It is widely believed that the Standard Model must be a low-energy approximation to a more general theory that does not have these fine-tuning “hierarchy” problems, and that the Large Hadron Collider (LHC) should uncover new physics associated with the more general theory [5]. One possible generalization of the Standard Model is Supersymmetry. In Wess-Zumino- type theories [4,6,7], the supersymmetry between bosons and fermions effectively regulates Higgs mass divergences, making these theories “natural” and applicable up to much higher energy scales [4]. Another possible generalization of the Standard Model is Technicolor [8-10]. Technicolor theories posit the existence of new interactions that create nonperturbative vacuum condensates that break the Electroweak symmetry and generate masses. These theories do not have any fundamental scalar fields (like a Higgs), so there are no “unnatural” quadratic divergences encountered during renormalization. However, Technicolor condensates are inherently nonperturbative, so they cannot be calculated directly. This paper explores a different mechanism for mass generation that could ultimately lead to new generalizations of the Standard Model. Like Technicolor, the model considered here has no scalar fields, so there are no quadratic divergences that require unnatural fine tuning. But instead of a condensate playing the role of the Higgs to generate masses, “bosonic” ghosts play that role. These ghosts have bosonic statistics (unlike normal Fadeev Popov ghosts), and they arise when fixing the gauge of certain supersymmetric gauge theories. Due to their statistics and the fact that they have zero spin, bosonic ghosts can have a vacuum expectation value (VEV) without violating Lorentz invariance. This VEV can lead to effective masses in the same way that the Higgs VEV does in the Standard Model, but without the troublesome quadratic divergences that plague the Higgs. The gauge theory explored in this paper is built around the supersymmetric group SU(2/3). Like SU(5), the fundamental representation of SU(2/3) is generated by 24 Hermitian 5x5 matrices. Like SU(5), 12 of SU(2/3)’s matrices form a U(1) x SU(2) x SU(3) subgroup of generators that correspond to bosonic gauge fields. But unlike SU(5), the remaining 12 matrices correspond to fermionic gauge fields. Throughout this paper, the 12 bosonic gauge directions will be referred to as being “even” and the 12 fermionic directions as “odd”. These labels are motivated by the commutation relations presented below. SU(2/3) is similar to the group SU(2/1), which was first presented in the literature in 1979 in an attempt to explain why the Weinberg angle was so close to 30 degrees. In [11,12] it was shown that if one added 2 anti-commuting Grassman dimensions to the usual four dimensions of spacetime, then the “odd” gauge fields with Grassman indices would be bosonic Lorentz scalars with regard to the usual four spacetime indices. As a result, they could behave like a Higgs field, developing a vacuum expectation value (VEV) and generating masses for the vector bosons. Later works pointed out a number of problems with this theory, including violations of the spin-statistics theorem and the fact that the “U(1)” part of the Lagrangian had the “wrong” sign [13]. This wrong sign made it impossible to mix the U(1) field with the diagonal SU(2) field in the usual way, and it also implied that the U(1) field either had negative kinetic energy or an indefinite metric that would prohibit the existence of asymptotic photons. Later, it was also pointed out that the “Weinberg angle” of the theory was actually 45 degrees [14], not 30 degrees as previously thought. Eventually SU(2/1) was mostly abandoned as a potential theory for the ElectroWeak interaction, although there is still some work being done in that area [15]. Recently, the negative kinetic energy of the U(1) field has led to the reintroduction into the literature of SU(2/1) as a possible theory for cosmological phantom energy [16]. This more recent work assumes only the four normal spacetime dimensions (no Grassman dimensions). It also formulates the theory with the odd (fermionic) gauge fields being spin-1 vectors. Problems with violation of the spin-statistics theorem are avoided by assuming that the fermionic, spin-1, odd fields are always bound in bosonic doublets. The formulation of SU(2/3) in this paper also involves just the normal four spacetime dimensions and vector fields for both the even and odd gauge directions. Fixing the gauge of the theory produces the usual fermionic ghosts associated with the bosonic gauge fields, but it also produces bosonic ghosts associated with the fermionic gauge fields. Like the bosonic fields of [11,12,15], these bosonic ghosts have the right spin and statistics to behave like Higgs bosons, but they do not require additional Grassman spacetime dimensions. Using the Antifield formalism [17,18], it is shown that one can pick a gauge with a Higgs-like potential for the bosonic ghosts that is minimized when they develop a VEV. This VEV produces vector boson masses and mixing that imply a “Weinberg angle” of 30 degrees. The Group SU(2/3) λ The group SU(2/3) is generated by 24 5x5 matrices A , where the index A runs from 0 to 23. For indices 1-23, the matrices are the same as SU(5) matrices, and they are grouped into blocks as follows: i, j,k ∈{}1− 3 λ ∈ SU (2) | λβ i α,β,γ ∈{}4 −15 , (1) λ | λ ∈ SU (3) β a a,b,c ∈{}16 − 23 where the index conventions used in this paper are also shown. In addition to those 23 matrices, there is also a diagonal U(1) matrix, λ = ()1 1 1 1 1 0 3 diag 2 ,2 , 3 , 3 , 3 . (2) All of the generators have vanishing supertraces, where the supertrace for this theory is defined by adding the first two diagonal entries of a matrix and subtracting the last three. The indices α,β ,γ ∈{}4 −15 are labeled as “odd”, and all other indices are “even”. Just as in SU(5), the “even” generators close in a U(1) x SU(2) x SU(3) subgroup. An SU(2/3) gauge transformation = ()λ ε ε takes the form U exp i A A , where A is a commuting variable for “even” indices and an anticommuting (Grassman) variable for “odd” indices. C Since SU(2/3) is a closed group, one can define structure constants f AB through the following equations: {}λ λ = C λ ∈{}− A , B if AB C for A and B 4 15 (both are “odd” indices) []λ λ = C λ ∉{}− A , B if AB C for A or B 4 15 (at least one index is “even”) . (3) The top relation shows that the anticommutator of two “odd” fields makes an “even” field. This, combined with the fact that the “even” generators close in their own subgroup, is the motivation for the “even” and “odd” naming convention. The covariant derivative and field strength tensor are given by: ≡ ∂ − Aλ Dµ µ igAµ A λ C ≡ [] g C Fµν i Dµ , Dν = λ ()∂ C − ∂ C + C A B g C µ Aν ν Aµ gf AB Aµ Aν . (4) The gauge invariant Lagrangian for the theory is given by a supertrace: = − 1 ()λ A λ Bµν L0 2 str A Fµν B F = − 1 0 0µν − 1 i iµν + 1 a aµν + 3 α α βµν 4 Fµν F 4 Fµν F 4 Fµν F 2 if0β Fµν F .